Universal quantifier

In a book "How to prove it" by velleman, universal quantifier is defined as follows;
To say that P(x) is true for every value of x in the universe of discourse U, we will write [itex]\forall[/itex]xP(x). This is read "For all x, P(x)". The symbol [itex]\forall[/itex] is called the universal quantifier.
And in this book, universe of discourse is defined as a set of all possible values for the variables relating with a statement.

Then, what I want to know is if under a special condition like that the universe of discourse of a statement is not a set, the universal quantifier can't be used to express such the statement.
For example, since there's no set which contains all ordinal numbers, I expect that a statement like "for all ordinal numbers x, P(x)" can't be expressed by using universal quantifier.
In a book " introduction to set theory" by karel hrbacek, the set of all natural numbers is
defined as [itex]\left\{[/itex] x : x[itex]\in[/itex] I for every inductive set I[itex]\right\}[/itex].
In this case, the universe of discourse of the statement is the set of all inductive set.
But , i can't convince of the existence of such a set of all inductive set.2. Relevant equations

3. The attempt at a solution
In some books described by NBG set theory, universal quantifier is just defined without the use of universe of discourse. I thought that this is because the notion class can be replaced with it. I mean, I thought that the definition of universal quatifier is such that for all x contained in a specified class, P(x).
I'm very confused about this notion and I doubt myself whether asking this question is meaningful or not....
I wanna know what is wrong in my argument and I ask you the exact definition of universal quantifier and also wanna know if the notion universe of discourse is needed in defining universal quantifier.